An Alternate to the Unified Distribution with Application to Breast Cancer Data
The Unified (θ) distribution appeared in , and in this short note we introduce a variant of this distribution. Further, a member of this family is shown to be significant in cancer modeling.
Bates distribution; Unified distribution; Cancer modeling
THE UNIFIED (θ) DISTRIBUTION
For x ∊(0,1) , the PDF is given by
Remark 3.1. For details on the derivation of this family, see 
THE NEW FAMILY
Let X be Bates on (0, 1) , and denote its PDF as f x (x,n) and consider
where k and ø are defined as in . Suppose Q=q, and consider =a+(b-a)J, where the random variable J has density q then
So, the CDF of Y is
By differentiation the PDF of Y is
So, if ø=1, then our density becomes
The class of all distributions with the above density is denoted A Unified (a, b, θ)
Remark 4.1. The families A Unified (a, b, θ) and Unified (θ) (of Section 1) coincide if and only if a=0 and b=1
THE UNIFIED (θ) GENERATED FAMILY OF DISTRIBUTIONS
The A Unified(a, b, θ) generated family of distributions has CDF
where 0 ≠θ ϵR, and K(x) is the CDF of some baseline distribution. Assuming the baseline distribution is Weibull with the following CDF
for x, g, h>0, then from the integral we have the following
Proposition 5.1. The CDF of the A Unified (0, 1, θ) Weibull family is given by
where x, g, h>0 and 0 ≠ ϵR
Obviously, the PDF can be obtained by differentiation. We write H AUW g h (θ, , ), if H is a A Unified (a, b, θ) Weibull random variable. The family in the above Proposition is a good fit to real life data as shown below. The MLE estimates are obtained using the software MATHEMATICA (Figure 1) .
In the present paper we introduced an alternate to the Unified distribution appearing in , and showed the Unified distribution is good in fitting real life data.
I am grateful to Oscar Alberto Quijano Acura of Concordia University, Montreal, Canada for pointing out an error in an earlier version of this paper.
- Oscar AQA (2019) The unified distribution. Journal of Statistical Distributions and Applications 6: 13.
- Bates distribution. Wikipedia, The Free Encyclopedia.
- Jayakumar K, Girish BM (2017) T-Transmuted X family of distributions. STATISTICA 77(3): 251-276.
Received date: November 19, 2019
Published date: November 25, 2019
Address for correspondence
Clement Boateng Ampadu; 31 Carrolton Road, Boston, MA 02132-6303, USA
©2019 Open Access Journal of Biomedical Science, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use. Open Access Journal of Biomedical Science is licensed under a Creative Commons Attribution 4.0 International License
How to cite this article
Clement BA. An Alternate to the Unified Distribution with Application Breast Cancer Data. 2019 - 1(2) OAJBS.ID.000117.